Periodically-driven quantum systems: Effective Hamiltonians and engineered gauge fields

Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating non-trivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micro-motion. This framework allows for the identification of driving schemes, based on general N-step modulations, which lead to configurations relevant for quantum simulation. In particular, we explore methods to generate synthetic spin-orbit couplings and magnetic fields in cold-atom setups.


I. INTRODUCTION
Realizing novel states of matter using controllable quantum systems constitutes a common interest, which connects various fields of condensed-matter physics. Two main routes are currently investigated to reach this goal. The first method consists in fabricating materials [1,2], or artificial materials [3][4][5][6][7][8][9][10], which present intrinsic effects that potentially give rise to interesting phases of matter. For instance, this is the case for topological insulating materials, which present large intrinsic spin-orbit couplings [1]. The second method, which is now commonly considered in the field of quantum simulation, consists in driving a system using external fields [11][12][13] or mechanical deformations [4,14] to generate synthetic, or effective, gauge structures. Formally, these driveninduced gauge fields enter an effective Hamiltonian, which captures the essential characteristics of the modulated system. This strategy exploits the fact that modulation schemes can be tailored in such a way that effective Hamiltonians reproduce the Hamiltonians of interesting static systems. Furthermore, the versatility of driving schemes might enable one to explore situations that remain unreachable in static fabricated systems.
In this work, we develop and explore a general framework that describes periodically-driven quantum systems, and which generalizes the formalism introduced by Rahav et al. in Ref. [57]. In contrast with the standard Floquet analysis [58,59] or the effective-Hamiltonian method presented by Avan et al. in Ref. [60], the present method clearly isolates and identifies the three main characteristics of modulated systems: (1) the effective Hamiltonian underlying the long-time dynamics; (2) the micro-motion; and (3) the effects associated with the initial phase of the modulation. These distinct effects will be largely illustrated in this work, based on different examples relevant for the quantum simulation of gauge structures, e.g. magnetic fields and spin-orbit couplings. Moreover, this work provides general formulas and methods, which can be easily exploited to identify wide families of promising driving schemes.
Before presenting the outline of the paper [Section I C], we briefly summarize some important notions related to driven quantum systems, based on basic illustrative examples.

A. Effective Hamiltonians and the micro-motion: Two simple illustrations
We start the discussion by presenting two very simple situations, which illustrate in a minimal manner the basic notions and effects encountered in the following of this work.

The Paul trap
This first illustrative and basic example consists in a particle moving in a modulated harmonic trap [61]. The Hamiltonian is taken in the form H(t) =Ĥ 0 +V cos(ωt) =p 2 2m + 1 2 mω 2 0x 2 cos(ωt), (1) where ω = 2π/T [resp. ω 0 ] denotes the modulation [resp. harmonic trap] frequency. The evolution operator after one period of the modulation is evaluated in Appendix A, and it readŝ arXiv:1404.4373v3 [cond-mat.quant-gas] 10 Jun 2015 expressing the fact that the particle effectively moves in a harmonic trap with frequency Ω = ω 2 0 / √ 2ω. An additional insight is provided by a classical treatment, in which one partitions the motion x(t) =x(t) + ξ(t) into a slow and a fast (micro-motion) component. As shown in Appendix A, this analysis shows that the effective harmonic potential with frequency Ω that rules the slow motionx(t) is equal to the average kinetic energy associated with the micro-motion: where . denotes the average over one period. This classical result illustrates the important role played by the micro-motion in modulated systems.

The modulated optical lattice
As a second example, we consider a modulated 1D lattice, treated in the single-band tight-binding approximation [62][63][64][65][66]. The Hamiltonian is taken in the form [see Appendix B] whereĤ 0 describes the nearest-neighbour hopping on the lattice, and where the operatorâ † j creates a particle at lattice site x = ja, and a is the lattice spacing. The effective Hamiltonian describing the slow motion of a particle moving on the modulated lattice can be derived exactly [see Appendix B], yielding the well-known renormalization of the hopping rate by a Bessel function of the first kindĤ eff = J 0 (κ/ω)Ĥ 0 ≈ 1 − This effect has been observed experimentally with cold atoms in optical lattices [63,65]. The micro-motion also plays an important role in this second example, where it is associated with large oscillations in quasimomentum space. Indeed, within the single-band approximation, a significant modification of the tunneling rate is found when the micro-motion oscillation is comparable to the width of the Brillouin zone.
We point out that the Paul trap and the modulated lattice share similar structures [see also Appendices A and B]: both systems are driven by a modulation of the formĤ 0 +V cos(ωt), and their effective Hamiltonians both contain a non-trivial term which is second order in the period T . In the present case of the modulated lattice, the term ∼ (κ/ω) 2 is the first non-trivial term of an infinite series, which can be truncated for κ/ω < 1, see Eq. (5).
B. The two-step modulation and the ambiguity inherent to the Trotter approach Motivated by the two simple examples described above, we consider a general quantum system described by a static Hamil-tonianĤ 0 , which is periodically driven by a repeated two-step sequence of the form whereV is some operator. For simplicity, we suppose that the duration of each step is T /2, where T = 2π/ω is the period of the driving sequence γ. Thus, the square-wave sequence γ is qualitatively equivalent to the smooth drivingĤ 0 +V cos(ωt) encountered in the two examples discussed above.
In the following of this work, the energy ω will be considered to be very large compared to all the energies present in the problem, justifying a perturbative treatment in (1/ω). The small dimensionless quantity associated with this expansion, Ω eff /ω, will be made explicit for the various physical problems encountered in the following Sections. Typically, Ω eff will be identified with the cyclotron frequency in the case of synthetic magnetism (see Section VI), or with the spin-orbit coupling strength (see Section VII); see also Section VIII A.
Starting in an initial state |ψ 0 at time t i = 0, the state at time t = N T (N ∈ N) is obtained through the evolution operator where we introduced a time-independent effective Hamiltonian, The product of two exponentials can be simplified through the Baker-Campbell-Hausdorff (BCH) formula, hereafter referred to as the Trotter expansion, yielding a simple expression for the effective "Trotter" Hamilto-nianĤ Importantly, the sign in front of the first order term depends on the starting pulse (Ĥ 0 +V orĤ 0 −V ) of the driving sequence (6), or equivalently, on the definition of the starting time t i : indeed, shifting the starting time t i → t i + (T /2), leads to the opposite term +i(T /4)[Ĥ 0 ,V ]. Thus, the first-order term arising from the Trotter expansion is sensitive to the initial phase ωt i of the driving. We emphasize that the sign ambiguity is different from the phase of the micro-motion sampling, which will be illustrated later in this work. Furthermore, we note that the first-order term in Eq. (10) can be eliminated by a unitary transformation indicating its trivial role in the effective Hamiltonian. Indeed, the long-time behavior of the system described by Eq. (7) can be expressed aŝ which indicates that the system first undergoes an initial kick S = e − iT 4 V [when the sequence γ is applied in this order], then evolves "freely" for a long time t = N T , and finally undergoes a final sudden kick (i.e. a micro-motion). From Eq. (11), we conclude that the first-order term in Eq. (10) cannot be exploited to modify the band structure ofĤ 0 , or equivalently, to generate nontrivial gauge structures (e.g. effective magnetic fields or spin-orbit couplings). This observation is in agreement with the effective Hamiltonians obtained for the Paul trap (2) and the modulated lattice (5), where the first non-trivial terms were found to be secondorder in the period T , see Sections I A 1-I A 2.
C. Outline of the paper The following of the text is structured as follows: • Section II presents the general formalism used to treat timedependent Hamiltonians. The method is then applied to the simple two-step modulation introduced in Eq. (6).
• Section III illustrates the impact of the initial phase of the modulation on long-time dynamics, based on a simple example. This Section highlights the importance of the "kick" operatorK(t) introduced in Section II.
• Section IV derives useful formulas for the effective Hamiltonian and kick operators in the general case of N -step modulations (N ∈ Z).
• Section V explores two specific classes of modulations, characterized by N = 4 different steps.
• Section VI applies the latter results to a modulation generating an effective magnetic field in two-dimensional systems. This sequence is explored both in the absence and in the presence of a lattice.
• Section VII proposes and explores several driving sequences realizing effective spin-orbit couplings in twodimensional spin-1/2 systems. These sequences are also analyzed in the absence and in the presence of a lattice.
• Section VIII is dedicated to general discussions and conclusions. This final part analyses the convergence of the perturbative approach introduced in Section II. It also briefly discusses the possibility to launch the modulation adiabatically. Finally, we present concluding remarks and outlooks.

A. The formalism
Having identified the subtleties proper to the analysis based on the BCH-Trotter formula in Section I B, we now consider an alternative approach inspired by Ref. [57]. Let us first rephrase the general problem based on our previous analysis. We act on an initial state |ψ 0 with a time-periodic Hamiltonian between times t i and t f , the period of the driving being T = 2π/ω. In Eq. (14), we explicitly Fourier expand the timedependent potential, to take higher harmonics into account. There are three distinct notions: 1. The initial phase of the Hamiltonian at time t i (i.e. ωt i mod 2π): the way the driving starts, namelyV (t i ), may have an important impact on the dynamics; 2. The evolution of the system between the interval ∆t = t f − t i , which can be arbitrary long, and during which the HamiltonianĤ(t) is applied; 3. The final phase of the Hamiltonian at time t f (i.e. ωt f modulo 2π): this final step describes the micro-motion.
These concepts were illustrated in Eq. (12), for the simple twostep sequence (6) presented in Section I B. In order to separate these three effects in a clear manner, we generalize the approach of Ref. [57] and re-express the evolution operator aŝ where we impose that: •Ĥ eff is a time-independent operator; •K(t) is a time-periodic operator,K(t + T ) =K(t), with zero average over one period; •Ĥ eff does not depend on the starting time t i , which can be realized by transferring all undesired terms into the "kick" operatorK(t i ). Similarly,Ĥ eff does not depend on the final time t f .
Following a perturbative expansion in powers of (1/ω), we obtain [see Appendix C] In Eq. (16), we have omitted the second-order terms that mix different harmonics, noting that these terms do not contribute in the situations presented in this work; the complete second-order terms contained inĤ eff andK(t) are presented in the Appendix C, see Eqs. (C10)-(C11). By construction, and in contrast to the Trotter approach, the expressions (16)-(17) constitute a strong basis to evaluate the relevance of periodic-driving schemes in view of realizing non-trivial and robust effects, such as non-zero effective magnetic fields. We conclude this Section by pointing out that effective Hamiltonians can also be obtained through Floquet theory [58]. However, as apparent in Refs. [58,59], there is a priori no natural constraint within Floquet theory that prevents the "Floquet" effective Hamiltonians to contain t i -dependent terms. A possible way to get rid of these terms in Floquet theory is to consider an adiabatic launching of the driving [64,65], such that the evolving state is constrained to remain in the same (principal) quasienergy multiplicity at all times (the multiplicity being well separated by the large energy ω). However, for the sake of generality and clarity, we will follow here the approach based on the partitionment (15) discussed in this Section, which provides an unambiguous and physically relevant definition for the effective Hamiltonian. Finally, we point out that the convergence of the perturbative expansion in powers of (1/ω) and leading to Eq. (16) is by no means guaranteed, as will be discussed later in Section VIII A.
B. Illustration of the formalism: back to the two-step sequence As a first illustration, let us apply the expressions (16)- (17) to the simple two-step sequence in Eq. (6). The Hamiltonian is given is the standard square-wave function. Expanding f (t) into its Fourier components, we obtain a simple expression for theV (j) operators introduced in Eq. (13).
such that the evolution operator is given by [Eq. (15)] in agreement with Eq. (12). Note that the amplitude of the initial kickK(t i ) is maximal at the initial time t i = 0, and that it is zero at time t i = T /4. In contrast with the Trotter analysis, the approach based on the partitionment (15) directly identifies: (a) the absence of first-order term in the effective HamiltonianĤ eff , (b) the initial kick produced by the op-eratorŜ = exp(iK(0)) = exp(−iTV /4), and (c) the micromotion exp(−iK(t)). We note that since the latter operator satisfies exp(−iK(N T )) = exp(−iK(0)) = exp(iTV /4) =Ŝ † , we exactly recover Eq. (12) for t = N T . The result in Eq. (18), which is associated with the two-step sequence (6), is to be compared with the smooth driving considered in Sections I A 1-I A 2, which is readily treated using Eq. (16).
We now apply the formalism to the two examples presented in Sections I A 1-I A 2: a. The Paul trap We readily recover the effective Hamiltonian in Eq. (2) by inserting the operators defined in Eq. (1) into Eq. (21). Furthermore, Eq. (22) provides an approximate expression for the micro-motion underlying the slow motion in the Paul trap [see also Appendix A for more details].
b. The modulated lattice Inserting the operators defined in Eq. (4) into Eqs. (21)- (22) yieldŝ where we indeed recover the first terms of the Bessel function expansion in Eq. (5). We note that the maximal amplitude of the kick associated with the micro-motion is given by exp , which corresponds to a translation in the Brillouin zone by an amount ∆k = κ/ω. We thus recover the fact that the modification of the hopping rate becomes appreciable when the micro-motion is comparable to the width of the Brillouin zone, ∆k ≈ π: indeed J 0 [(κ/ω) = π] ≈ −1/3 is very close to the minimal value of the Bessel function, and thus corresponds to a dramatic change in the tunneling rate (the maximal value of the Bessel function is J 0 (0) = 1, which corresponds to the standard hopping rate in the absence of shaking). The result in Eq. (23) stems from the perturbative expansion in powers of (1/ω). However, the formalism presented in Section II A also allows for an exact treatment of the modulated-lattice problem. The full derivation is given in Appendix E, where we recover the Bessel-renormalized-hopping result of Eq. (5). Moreover, this derivation provides an exact form for the kick operator, see Eq. (E10). The latter expression indicates that the maximal amplitude of the kick operator is given byK =xκ/ω, which is precisely the result discussed above based on the perturbative treatment.

III. LAUNCHING THE DRIVING: ILLUSTRATION OF THE INITIAL KICK
In the last Section, we introduced the partitionment of the evolution operator which highlights the fact that the system undergoes an initial kick exp[iK(t i )] before evolving according to the (time-independent) effective Hamiltonian. This initial kick depends on the launching time of the sequence t i , through Eq. (17), and it can have a great impact on long-time dynamics. It is the aim of this Section to illustrate this effect, based on a basic but enlightening example. Consider a particle driven by a uniform force that alternates its sign in a pulsed manner: +F, −F, . . . This system obeys the two-step sequence in Eq. (6) withĤ 0 =p 2 /2m,V = −Fx. The effective Hamiltonian and kick operators are readily obtained through Eqs. (18)- (19), yieldinĝ The initial kick operator isK(t i ) =xF T /4 for t i = 0,K(t i ) = −xF T /4 for t i = T /2, and its effect is thus to modify the initial mean velocity v(t i ) → v(t i ) ± F T /4m before the long-time free evolution. We emphasize that the initial kick operator is zero when starting the sequence at time t i = ±T /4.
Further insight is provided by computing the evolution operator exactly, at time t = T . This readŝ where the gauge potential is given by A(t i ) = ±F T /4 whether the starting time is t i = 0 or t i = T /2, respectively. We note that the effective HamiltonianH eff (t i ) is related to the t iindependent effective HamiltonianĤ eff in Eq. Treating the basic mechanics problem associated withH eff (t i ) semi-classically, we recover that the initial kick modifies the initial velocity v(t i ) → v(t i ) + A/m. Hence, for an arbitrary v(t i ), the position of the particle x(t T ) will significantly depend on whether the pulse sequence has started with a pulse +V [t i = 0] or −V [t i = T /2]: the dynamics is strongly sensitive to the initial phase of the modulation. In Fig. 1, we compare these predictions to the real (classical) dynamics of the pulse-driven particle. This figure illustrates the sensitivity to the initial phase of the driving, but also, the effects of micro-motion present at all times for all configurations.
In general, the effects related to the initial phase of the modulation may remain important in more sophisticated driven systems. The sensitivity to the initial phase will be further illustrated in the context of driven-induced Rashba spin-orbit couplings in Section VII [see Fig. 5].

IV. MULTISTEP SEQUENCES
Going beyond the two-step driving sequence in Eq. (6) potentially increases the possibility to engineer interesting effective Hamiltonians and gauge structures. We now derive the effective Hamiltonian for the general situation where the pulse sequence is characterized by the repeated N -step sequence where theV m 's are arbitrary operators. In the following, we consider that the duration of each step is τ = T /N , where T is the driving period, and we further impose that N m=1V m = 0. The general driving sequence γ N is illustrated in Fig. 2. Note that γ N reduces to the simple sequence (6) of Section I B for N = 2 and The general Hamiltonian corresponding to the pulse sequence (27) is written aŝ -1000 0 100 200 0 1000 Figure 1. Sensitivity to the initial phase of the driving. (brown) Classical dynamics of a particle driven by a repeating pulse sequence {+F, −F }, where F > 0 is a uniform force; the period is T = 20 and the particle is initially at rest v(ti) = 0 at x(ti) = 0. (red) Same but considering the sequence {−F, +F }, i.e. shifting the starting time ti → ti + T /2. (blue) Same but starting the sequence with −F during a time T /4, and then repeating the sequence {+F, −F }, i.e. shifting the starting time ti → ti − T /4. Note that the initial kickK(ti) is inhibited in the latter case, while it is opposite in the two former cases [Eq. (26)]. In all the plots, the particle depicts a small micro-motion, captured byK(t) in Eq. (26). The brown and red curves highlight the great sensitivity to the initial phase of the modulation . where the last line of Eq. (28) provides the Fourier series of the square functions f m (t). In order to apply Eqs. (16) and (17), we expand the Hamiltonian in terms of the harmonicŝ where we used the Fourier series in Eq. (28). The effective Hamiltonian and the initial-kick operatorK(0) are then given by the general expressions [see Appendix F] where C m,n = N 2 + m − n and D m, We have again omitted the harmonic-mixing terms given in Eq.
(C10), which do not contribute for the sequences considered here. The result in Eq. (31) clearly highlights the fact that the initial kick K(0) depends on the way the pulse sequence starts, whereas the effective HamiltonianĤ eff is independent of this choice: shifting the pulse sequence, namely redefining the operatorsV m →V m+p , with p ∈ Z, results in a change inK(0) but leavesĤ eff invariant.
We have applied the formula (31) to general sequences with N = 3 and N = 4 different steps, and we present the associated results in Eqs. (G1)-(G2) in Appendix G. In contrast with the case N = 2, we find that sequences with N = 3 or N = 4 steps can potentially lead to non-trivial effects that are first order in (1/ω). We point out that the scheme proposed by Kitagawa et al. [49] to realize the Haldane model using a modulated honeycomb lattice corresponds to the case N = 3. Moreover, the model of Refs. [51,55], which features topological "Floquet" edge states, corresponds to the case N = 5. In the following of this work, we will further explore and illustrate the case N = 4, with a view to creating synthetic magnetic fields and spin-orbit couplings with cold atoms (see also [15,25,26]).

V. SEQUENCES WITH N = 4 STEPS
Motivated by the importance and versatility of four-step sequences to generate non-trivial effective potentials and gauge structures, we now explore two specific examples of such sequences that lead to different effects. The following paragraphs will constitute a useful guide for the applications presented in Sections VI and VII.
A. The class of sequences α Let us first consider the following four-step sequence which corresponds to γ 4 in Eq. (27) We now apply the formula (31) [see also Eq. (G2)] and we obtain where the expression forK(0) was obtained for a sequence α starting with the pulse +Â (and we remind that the kick opera-torK(0) depends on this choice). The result in Eq. (33) shows that the driving schemes belonging to the class α can generate a combination of first-order and second-order terms, which might potentially lead to interesting observable effects, see Sections VI and VII. In particular, we note that the scheme of Ref. [15] to generate synthetic magnetic flux in optical lattices belongs to this class, see Section VI. Finally, we note that the pulse sequence α can be approximated by the smooth drivingV (t) =Â cos(ωt)+B sin(ωt). In this case, a direct evaluation of Eq. (16) yieldŝ which is indeed approximatively equal to the effective Hamiltonian (33) associated with the pulsed system.

B. The class of sequences β
We now consider an apparently similar four-step sequence which corresponds to γ 4 in Eq. (27) whereK(0) corresponds to a sequence β starting with the pulse +Â. The result in Eq. (35) emphasizes two major differences between the α and the β classes: (a) the effective Hamiltonian associated with the β sequence does not contain any first-order term, and in this sense, this N = 4 sequence resembles the case N = 2; (b) the β sequence generates additional second-order terms that mix the pulsed operatorsÂ andB, i.e. terms of the form , which are not present in the class α [Eq. (33)].
As illustrated in Section VII, the schemes of Refs. [25,26] to generate spin-orbit couplings in cold gases can be expressed in the form of sequences β. In this Section, we apply the results of Section V to generate synthetic magnetic fields in one-component atomic gases, using a four-step sequence of type α, see Eq. (32).

A. Without a lattice
We first consider atoms moving in two-dimensional free space, such thatĤ 0 = (p 2 x +p 2 y )/2m. Inspired by Ref. [15], we drive the system with a pulse sequence α, see Eq. (32), with the operatorŝ A = (p 2 x −p 2 y )/2m andB = κxŷ. We will comment later on the possibility to implement such a scheme practically. Over a period, the evolution is thus given by the sequence which consists in repeatedly allowing for the movement in a pulsed and directional manner, while subjecting the cloud to an alternating quadrupolar field. The corresponding effective Hamiltonian is given by Eq. (33), which yields, up to second order (1/ω 2 ), which corresponds to the realization of a perpendicular and uniform synthetic magnetic field We point out that the second-order corrections in Eq. (33) lead to a harmonic confinement, which dominates over the centrifugal force generated by the first-order term, resulting in an overall trapping potential in Eq. (37). Defining the cyclotron frequency ω c = B/m, we obtain the ratio between the confinement and cyclotron frequencies The induced confinement is a special feature of the driving scheme (36). It has a significant impact on the dynamics, which illustrates the fact that the perturbative expansion in (1/ω) should not be limited, in typical applications, to its first-order terms [see Section VIII A for a more detailed discussion].

B. With optical lattices
A similar scheme can be applied to cold atoms in optical lattices, where a uniform synthetic magnetic field would provide a platform to simulate the Hofstadter model [30,32,33,67,68].
Here, we suppose that the atoms evolve within a two-dimensional optical square lattice and that their dynamics is well captured by a single-band tight-binding description. The static Hamiltonian is thus taken in the form where J is the hopping amplitude,â † m,n creates a particle at lattice site x = (ma, na), and where a is the lattice spacing. In Eq. (41), we introduced the notationp 2 x,y /2m * to denote hopping along the (x, y) directions, and also, the effective mass m * = 1/(2Ja 2 ). In the following of this work, any operator denotedŌ will be defined on a lattice, with the convention thatŌ →Ô in the continuum limit [see Appendix H].
We now apply the pulse sequence in Eq. (36) to the lattice system, by substitutingp 2 x,y /2m →p 2 x,y /2m * andxŷ →xȳ. The lattice analogue of the sequence (36), which was originally introduced in Ref. [15], now involves pulsed directional hoppings on a lattice, and it can thus be realized using optical-lattice technologies. In this lattice framework, the effective Hamiltonian in Eq. (33) yieldŝ ,nâ m,n , where we introduced the "flux" Φ = a 2 κ/8ω, and used the commutators presented in Appendix H 1. In the small flux regime Φ 1, we obtain the Hofstadter Hamiltonian [30] H where the additional term acts as a harmonic confinement in the continuum limit, with frequency ω h = 2πΦJ 5/3. Noting that Φ is the number of (synthetic) magnetic flux quanta per unit cell [30], and denoting the cyclotron frequency ω c = B/m * , we recover the free-space results (38)-(39), namely which validates the analogy between the free-space and lattice systems. In the present perturbative framework, the flux is limited to Φ 1; however, a partial resummation of the series (42), similar to that of Section VII D, allows one to extend the flux range to Φ ∼ 1, see Ref. [15].
The lattice system is convenient for physical implementation, since optical lattices offer a platform to activate and deactivate the hopping terms in a controllable manner, e.g., by simply varying the lattice depths in a directional way. Besides, the lattice configuration described in this Section also allows for direct numerical simulations of the Schrödinger equation (the lattice discretization being physical). We have performed two types of simulations illustrating: (a) the dynamics of a gaussian wave packet subjected to the pulse sequence (36), and (b) the dynamics of the same wave packet evolving according to the effective Hamiltonian (42) with Φ = a 2 κ/8ω. In Fig. 3, we show the center-of-mass dynamics of a wave packet initially prepared around x(0) = 0 with a non-zero group velocity v g (0) = v1 y , v > 0. The non-trivial dynamics associated with the effective Hamiltonian (42)  x(0) Figure 3. Comparison between the dynamics of the driven system following the protocol in Eq. (36) (red and blue curves), and the dynamics predicted by the effective Hamiltonian (42) (purple dotted curve). Shown is the center-of-mass trajectory in the x − y plane for time t ∈ [0, 100](1/J). For all simulations, a gaussian wave packet is initially prepared around x(0) = 0 with a non-zero group velocity along the +y direction. The blue and red curves correspond to different initial phases of the driving sequence: the blue (resp. red) curve was obtained by starting the sequence (36) withĤ0 +Â =p 2 x /m * (resp.Ĥ0 −Â =p 2 y /m * ). We set the values κ = 10 and T = π/80(1/J), such as to fulfill the low-flux condition Φ = 1/128 1.
potential with frequency ω h . Moreover, Fig. 3 compares the trajectories resulting from a different choice of the initial phase of the modulation: the almost identical red and blue trajectories highlight the robustness of this scheme against perturbations in the driving's initial conditions, and incidentally, it shows the negligible role played by the initial kickK(t i ) in this example. Similarly, we also note that the micro-motion is negligible in real space. The numerical results presented in Fig. 3 confirm that the driving sequence (36), introduced in Ref. [15], produces an effective magnetic field in the optical-lattice setup; moreover, it illustrates the relevant effects associated with second-order (1/ω 2 ) corrections, which will be present even at low flux Φ 1. Moreover, we point out that adding terms in the static HamiltonianĤ 0 , for instance to further control the confinement of the gas or to combine several effects, should be treated with care, as these extra terms will potentially contribute to second-order corrections through the commutators In this Section, we investigate the possibility to generate spinorbit coupling (SOC) terms µν α µνpµĴν in a cold-atom gas, whereĴ denotes the spin operator associated with the atoms,p is the momentum and α µν are some coefficients. Here, for the sake of simplicity, the focus will be set on the case of two-dimensional and spin-1/2 Rashba SOC, which is given byĤ R = λ Rp ·σ. In the following, we will also encounter another SOC term,Ĥ Lσ = Ω SOLzσz , the so-called "intrinsic" or "helical" SOC, which is responsible for the quantum spin Hall effect in topological insulators [69,70]. The combination of both terms, e.g. as in the Kane-Mele model [70], will be referred to as the "helical-Rashba" configuration.
Before presenting different schemes leading to SOC terms, we point out a subtlety associated with Rashba spin-orbit coupled systems, which arises when considering a perturbative treatment in powers of λ R . ConsiderĤ 0 the free Hamiltonian in 2D space, and let us perform the following unitary transformation where we introduced the short notation O(λ R ) ≡ O(mλ R L), and L is the system's length. The result in Eq. (45) shows that -up to third order in λ R -any model combining the RashbaĤ R and helicalĤ Lσ SOC terms with the weight ratio Ω SO /λ R = mλ R , is equivalent to a trivial system described by the free Hamiltonian H 0 [26]. This observation highlights the importance of evaluating the SOC terms up to second order in λ R , to identify the schemes producing genuinely non-trivial spin-orbit effects. Moreover, in this perturbative framework, Eq. (45) indicates that the (firstorder) Rashba SOC HamiltonianĤ =Ĥ 0 +Ĥ R is equivalent to the (second-order) helical SOC HamiltonianĤ =Ĥ 0 +Ĥ Lσ , with Ω SO = mλ 2 R . Finally, we point out that the expansion in powers of mλ R L introduced in Eq. (45) should be handled with care for two main reasons. First, it should not be mistaken with the (1/ω) expansion stemming from the effective-Hamiltonian formalism of Section II A, which is typically characterized by the small dimensionless quantity Ω SO /ω. Second, the ground-states of the Rashba SOC Hamiltonianp 2 /2m + λ Rp ·σ are situated along the "Rashba ring" at p = p R = mλ R [i.e. the bottom of the mexican hat dispersion [12,13]], so that probing this region of the dispersion relation requires to preparing states with ∆p mλ R ; in this regime the expansion in Eq. (45) becomes problematic as mλ R ∆x 1/2, which is imposed by Heisenberg inequality.
B. Generating spin-orbit couplings with the α sequence

The helical-Rashba scheme
Inspired by the result presented in Section VI A, we propose a scheme to realize spin-orbit couplings, based on the four-step sequence α in Eq. (32). Considering the operatorŝ the time evolution of the driven system is characterized by the repeated sequence The general expression for the effective Hamiltonian (33) then yieldsĤ providing a "helical-Rashba" configuration. In this perturbative approach, we note that one can eliminate the Rashba or the helical SOC term in Eq. (48) via a unitary transformation, but not both since Ω SO = mλ 2 R [see the previous Section VII A]. The kick operatorK(t) is obtained from Eq. (17); in particular, the initial kick at t i = 0 is given by [Eq. (33)] In direct analogy with the discussion presented in Section III, we find that this initial kick can profoundly alter the dynamics of wave packets in the strong SOC regime, where κ/ω ∼ λ R is large. In this regime, it is thus desirable to launch the dynamics at a subsequent time t i = 3T /8, in which casê no longer depends on the parameter κ/ω. In the same spirit as in Section VI B, we now consider the lattice analogue of the driven system characterized by the pulse sequence (47). The corresponding tight-binding operators and commutators are presented in Appendix H 2. Similarly as in Section VI B, this scheme involves a combination of pulsed directional hoppings on the lattice and oscillating quadrupole fields. We obtain that the corresponding effective HamiltonianĤ eff reproduces the helical-Rashba Hamiltonian in Eq. (48), after substituting m by the effective mass m * = 1/(2Ja 2 ) and taking the continuum limit. A specificity of the lattice framework is that the second-order contributions also lead to a renormalization of the hopping amplitude J → J(1 − η 2 ), where η = (aπκ/4 √ 3ω); this is due to the fact that [[p 2 x ,x],x] =p 2 x a 2 in the lattice formulation [Appendix H 2]. We now demonstrate that the dynamics of the pulse sequence (47) is well captured by the predictions of the effective-Hamiltonian formalism. We consider that the system is initially prepared in a gaussian wave packet with non-zero group velocity along the +x direction and spin component σ = +. Figure  4 compares the dynamics generated by the pulse sequence (47) with the one associated with the evolution operator in Eq. (15), withĤ eff andK(t) given by Eqs. (17)-(48)-(G4). We show in Figs. 4(a)-(b) the spin populations as a function of time: using a time step ∆t < T , we observe a wide micro-motion in spin space, which is very well captured by the kick operatorK(t) of the effective model. Figure 4(c) compares the center-of-mass motion of the real and effective evolution operators. The curved trajectory, together with the evolution of the spin populations, signals the presence of the effective SOC generated by the driving. In agreement with the discussion of Section VII A, we recover the fact that it is necessary to evaluate the effective Hamiltonian up to (at least) second order in 1/ω to reach a good agreement with the dynamics of the real pulsed system. Adding third order corrections to the effective Hamiltonian [Eq. (G4) in Appendix G] leads to an even better agreement. Finally, similarly as in Fig. 3, we find that the micro-motion is small in real space.

The pure Dirac regime
Interestingly, Eq. (48) suggests that inhibiting the effect ofĤ 0 during the evolution will lead to a pure Dirac system , which, in principle, can be realized in an optical-lattice setup. Hence, controlling the static HamiltonianĤ 0 , which is conceivable in a lattice framework, offers the possibility to tune the first-order effective energy spectrum, but also, to annihilate the second-order contributions to the SOC effects. We note that the pulsed operatorsÂ andB in Eq. (46) could also be slightly modified to reach other regimes of interest.

Adding terms to the static HamiltonianĤ0: a route towards topological superfluids and topological insulators
The scheme based on the driving sequence (46)-(47) allows for adding a Zeeman termĤ Z = λ Zσz to the effective Hamiltonian (48), whose association with the Rashba SOC term could be useful for the quantum simulation of topological superfluids [71]. This could be simply realized by subjecting the system to a static Zeeman field,Ĥ 0 →Ĥ 0 +Ĥ Z ; this will not perturb the first-order Rashba termĤ R ∼ [Â,B] in Eq. (48), but will add an extra secondorder term which corresponds to a spin-dependent harmonic potential with frequency Ω Z = 2 √ λ Z Ω SO . The survival of the topological superfluid phase in the presence of the additional harmonic potential constitutes an interesting open question. In general, we note that any driving scheme aiming to produce Rashba SOC typically presents this potential drawback, namely, the fact that the additional Zeeman term will necessarily generate additional (possibly spoiling) effects.
Adding terms to the static HamiltonianĤ 0 could also be envisaged to generate systems exhibiting the quantum anomalous Hall (AQH) effect [56,72], the so-called Chern insulators. For instance, the AQH model of Ref. [72] could be realized by considering the driving sequence (46)-(47) on a square lattice, but replacing the static HamiltonianĤ 0 →Ĥ Z + (p 2 /2m)σ z , where the last term corresponds to a spin-dependent hopping term on the lattice. Another possibility would be to drive lattice systems with more complex geometries (e.g. honeycomb lattice), where the association of Rashba and Zeeman terms directly leads to Chern insulating phases [73,74].

The xy scheme
Finally, we introduce a second scheme based on the α sequence (32), which features the standard static HamiltonianĤ 0 =p 2 /2m and the pulsed operatorŝ The repeated driving sequence is similar to (47), reading so that this scheme is based on a more regular sequence involving pulsed directional motion and magnetic fields. The effective Hamiltonian (33) then readŝ Here, the second-order terms do not contribute to the effective Hamiltonian, but the first-order terms include a spin-dependent hyperbolic potential. Although the "xy" term is potentially problematic, especially in the strong SOC regime κ/ω , we will show in Section VII D that this pulse sequence allows for an almost exact treatment; incidentally, we will see that the "xy" term can be inhibited in the lattice framework through a fine tuning of the driving parameters.
where τ T ≈T , and where the operators are explicitly given byĤ Noting that the sequence (53) essentially features four non-trivial steps (with pulses ±Â and ±B), we find that it can be qualitatively described by the associated four-step sequence β in Eq. (34). The driving sequence studied in this Section, hereafter referred to as the "XA" scheme, will thus be taken of the form β, The corresponding effective Hamiltonian and initial kick operators then read [Eq. (35)] where we have introduced the effective HamiltonianĤ T eff . As anticipated at the end of Section VII A, Eq. (57) implicitly contains two intertwined perturbative expansions: the expansion in powers of (Ω SO /ω) inherent toĤ eff andK(t) [Section II A], and the expansion in powers of (mλ R L) introduced in Eq. (45).

D. Quasi-exact treatments on special cases
In this Section, we show that specific driving schemes leading to SOC benefit from the fact that they can be treated almost exactly. Such an approach is useful, as it allows to go beyond the perturbative treatment of Section II A, which has been considered until now to evaluate the evolution operator. Actually, we already encountered such a scheme in our study of the oscillating force in Section III.

The XA scheme
Following Refs. [25,26], the time-evolution operator associated with the sequence (55) can be conveniently partitioned aŝ U (T ) =Û yÛx , where each of the two subsequenceŝ can be treated exactly. The calculations presented in Appendix I yield the exact result [see also Refs. [25,26]] where λ R = πκ/8mω. Finally, the evolution operator after one period is obtained by using the Trotter expansion to the lowest order, exp A exp B ≈ exp(A + B), readinĝ We thus recover the result in Eq. (57), with the notable difference that the quasi-exact treatment leads to a partial resummation of the infinite series inherent to Eq. (57): indeed, the dimensionless parameter mλ R L no longer plays any role in the expression for the evolution operator in Eq. (60). Before discussing the result in Eq. (60) any further, we derive its lattice analogue by substituting the operators of the "XA" sequence (55) by their lattice counterparts [Appendix H 2]. Following the computations presented in Appendix I, we obtain the effective Hamiltonian up to first order in (Ω SO /ω) 2 , where λ R = πκ/8m * ω and m * = 1/2Ja 2 is the effective mass. Hence, we recover the result in Eq. (60) for weak driving λ R < aJ/2 and by taking the continuum limit. However, in the lattice framework, the maximum value of the effective Rashba SOC strength is limited: Using Eq. (61), we find that the ratio Rashba/hopping is maximized for λ R = π/4am * = (π/2)aJ: Note that the appearance of sinc functions in Eq. (61) is a characteristic of lattice systems driven by square-wave modulations [65].
The quasi-exact method presented in this Section allows to partially resum the infinite series contained inĤ eff andK(t) [Eq. (57)]. However, we stress that the evolution operators and the associated effective Hamiltonians in Eqs. (60)-(61) depend on the initial phase of the modulation, in direct analogy with the situation presented in Sections I B and III: the analysis performed in this Section imposes that the β sequence exactly starts with the pulse +Â and ends with the pulse −B. Any deviation in the initial phase will alter the evolution operator in Eqs. (60)- (61), and potentially, the long-time dynamics. Indeed, suppose that the launching time is shifted t i = 0 → −T /4, so that the β sequence starts with the pulse −B instead of +Â: the system will first undergo a kick before evolving according to the Rashba Hamiltonian in Eqs. (60)- (61). Note that δp = 4p R , where p R is the radius of the Rashba ring along which the ground-states are situated. Hence, this initial kick, which modifies the group velocity and spin structure of the prepared system, typically has an impact on long-time dynamics [in direct analogy with Fig. 1]. We illustrate this sensitivity to the initial phase of the driving in Fig. 5, which shows the time-evolved density of a lattice system driven by the sequence (55). Figure 5(a) shows the initial wave packet in real space, with mean position x = 0 and momentum k = 0. The width of the gaussian satisfies ∆k m * λ R in k-space, namely, the wave packet is prepared such as to probe the Dirac dispersion relation around k = 0 [i.e. within the Rashba ring]. Figure 5 63), the cloud undergoes a sudden kick along the y direction, which ejects the initial momentum distribution out of the Rashba ring, before evolving according to the Rashba Hamiltonian (61); hence, changing the initial phase of the modulation results in a highly anisotropic dynamics that no longer probes the Dirac dispersion relation proper to the Rashba Hamiltonian (61).
We also note that the treatment considered in this Section implies that the system is probed stroboscopically (t = N T ), and in this sense, it does not describe the micro-motion associated with the driving. More importantly, we point out that the exact treatment leading to Eq. (59) cannot be performed when adding Pauli matrices into the static Hamiltonian (e.g. an extra Zeeman term). Finally, we point out that a similar quasi-exact treatment was considered in Ref. [15], for the one-component driven lattice discussed in Section VI.

The xy scheme
The xy scheme introduced in Section VII B 4 can also be treated in an almost-exact manner, which leads to a partial resummation of the series in Eq. (52). The method differs from the one presented in the previous Section VII D 1 for the XA scheme, and (2) the driving parameter κ and the period T should satisfy the condition κT = 4π/a, or equally, using our previous definition, λ R = πκ/8m * ω = (π/2)aJ. We note that this particular value was already discussed below Eq. (61), where it corresponded to the regime where the effective Rashba SOC was maximized on the lattice. In the following, we assume that these two conditions are satisfied.
Here, in contrast with the analysis performed in the previous Section VII D 1, we split the evolution operatorŪ (T ) associated with the sequence (51) into its four primitive parts, and we analyze each operatorŪ ±A,B separately. Using the Zassenhaus formula [75], we find factorized expressions for the individual operators [see Appendix J for details] where λ * R was defined in Eq. (62). Finally, using Eq. (H6), and applying the Trotter expansion to minimal order, we find which is precisely the Rashba Hamiltonian in Eq. (62) up to the sign change λ * R → −λ * R (i.e. a gauge transformation). Interestingly, this shows that the xy scheme in Eq. (51) is equivalent to the XA scheme in Eq. (55), in the limit λ R → (π/2)aJ where the Rashba SOC is maximized. Furthermore, Eq. (66) shows that the hyperbolic potential, which is predicted by the perturbative treatment [Eq. (52)] for the lattice-free case, totally disappears in this special regime; this surprising result is due to the underlying lattice structure [Appendix J] .
We show in Fig. 6 the perfect agreement between the dynamics predicted by the effective Hamiltonian (66) and the real dynamics of the pulsed lattice system. One should note that this agreement is only valid in the special regime where λ R ≈ (π/2)aJ: the spoiling effects associated with the effective hyperbolic potential [Eq. (52)] become appreciable as soon as ∆λ R ∼ 1%. Finally, we stress that the analysis leading to Eq. (66) cannot be performed when the static HamiltonianĤ 0 features Pauli matrices (e.g. a Zeeman term).

VIII. DISCUSSIONS AND CONCLUSIONS
A. Convergence of the (1/ω) expansion As already pointed out in Ref. [57], one cannot affirm that the perturbative expansion proper to the formalism of Section II A converges in general. Indeed, the small dimensionless parameter associated with the 1/ω-expansion that leads to the effective HamiltonianĤ eff in Eq. (16) can only be determined a posteriori, on a case-by-case basis.

Illustration of the convergence issue
Let us illustrate this fact based on the driven system discussed in Section VI A. Using Eq. (16), we obtained the second-order effective Hamiltonian in Eq. (37), which we now decompose in terms of the 1/ω expansion: where Ω ∼ κ/mω and ω 2 conf ∼ Ω 2 . At this point, no element justifies the fact that the perturbative expansion should be carried up to second order. In order to simplify the present discussion, we slightly modify the sequence in Eq. (36) so that the Hamiltonian in Eq. (67) can be recast in the familiar form (68) which is easily done by changing the prefactor in front of the pulse operatorÂ ∼p 2 x −p 2 y entering the sequence in Eq. (36). Having obtained the effective Hamiltonian in Eq. (68) up to second-order in 1/ω, one can determine its general characteristics: for instance, let us focus on its ground state, which is the standard lowest Landau level (LLL) with cyclotron frequency Ω. This ground-state is characterized by the cyclotron radius r 0 ∼ 1/ √ mΩ, such that the typical momentum associated with this state is p 0 ∼ √ mΩ ∼ κ/ω. Hence, as far as the LLL is concerned, we find that all the terms in Eq. (67) are of the same order,Ĥ 0,1,2 ∼ Ω ∼ κ/mω, which indicates that the perturbative expansion leading to Eq. (67) should necessarily be undertaken up to second-order (included). Building on this result, we now evaluate the third-order terms, which have been neglected up to now: including the pulse operatorŝ A ∼p 2 x −p 2 y andB ∼xŷ into Eqs. (G3)-(G4), we find a cancellation of all the third order terms. We thus need to evaluate the fourth-order terms, which are necessarily of the form whereQ = (Ĥ 0 ,Â) ∼ p 2 /m. In the LLL, these fourth-order terms are all of order Ω 2 /ω or Ω 3 /ω 2 . Hence, we have identified that Ω/ω ∼ κ/mω 2 constitutes the relevant dimensionless parameter in the problem : the perturbative expansion leading to the effective Hamiltonian in Eq. (67) can indeed be safely limited to the second-order as long as Ω ω, namely when κ mω 2 .
Based on this example, we propose a guideline that should be followed in order to validate the convergence of the perturbative expansion: 1. Compute the effective HamiltonianĤ eff up to some order 1/ω n , using the formalism presented in Section II A; 2. Determine the interesting characteristics associated witĥ H eff , e.g. based on its eigenstates or its dispersion relation; 3. Evaluate the (n + 1)-order term and identify the condition according to which this term can (possibly) be neglected. This condition defines the small dimensionless parameter of the problem, or equally, a region in the parameter space, out of which higher order terms could come into play.

The perturbative approach revisited
We emphasize an important aspect related to time-dependent HamiltoniansĤ(t) =Ĥ 0 + m f m (ωt)V m , which is the fact that the operatorsĤ 0 ,V m are generally not independent with respect to the modulation frequency ω. For instance, in the example discussed above in Section VIII A 1, we found thatĤ 0 ∼Â ∼ Ω and B = κxŷ ∼ ω, based on the LLL characteristics [with Ω ω]. The fact that some driving operators may be proportional to the frequencyV m ∼ ω, which can only be determined a posteriori according to the guideline established above, seems problematic when building a perturbative expansion in powers of 1/ω [Section II A]. To treat this seemingly pathological situation, we propose an alternative perturbative approach in Appendix K, which is specifically dedicated to the general time-dependent problem where the functions f and g are assumed to be time-periodic with a zero mean value over one period T = 2π/ω. Applying this alternative method to lowest order in 1/ω provides a compact form for the effective HamiltonianĤ eff =Ĥ 0 where ζ(t) = (1/T ) T 0 ζ(t)dt denotes the mean value over one period, and where G(t) = ω t g(τ )dτ satisfies G(t) = 0.
Importantly, the formula in Eq. (70) potentially allows for a partial resummation of the perturbative expansion stemming from the formalism introduced in Section II A. To illustrate this point, we again consider the driven system in Section VI A, for which we found thatB ∼ ω. SubstitutingÂ →Â = (p 2 x −p 2 y )/2m and ωB →B = κxŷ into Eq. (70), and computing the time-averages associated with the α pulse sequence [Appendix K], we recover the effective Hamiltonian in Eq. (37). We find that only a few terms in Eq. (70) have a non-zero contribution, and we stress that, in contrast with the result (37) obtained using the formalism of Section II A, the present result based on the formula (70) guarantees the convergence of the perturbative expansion. Indeed, one readily verifies that all the terms that have been identified at this order of the computations [Eq. (70)] are of the same order Ω ∼ κ/mω [see Appendix K for more details].

B. Adiabatic launching
In this work, we assumed that the periodic modulation that drives the system is launched abruptly at some initial time t i . In Section III, we demonstrated that the long-time dynamics could strongly depend on this choice [Figs. 1 and 5]. Hence, after a long time t T , the system "remembers" the initial phase of the modulation, or equivalently, the very first pulse that was activated [e.g. +Â or −B in Fig. 5]. This sensitivity to the initial phase constitutes an important issue with regards to experimental implementation, where the phase is controlled with a certain uncertainty. One way to "erase" the memory of the system is to ramp up the modulationĤ(t) ==Ĥ 0 + λ(t)V (t), where λ(t) = 0 → 1 is turned on very smoothly. This "adiabatic launching" will effectively annihilate any effect associated with the initial kickK(t i ) in Eq. (15). We observe that the time-scales of such memory-eraser ramps depend on the scheme under scrutiny; for example, we find that the ramping time needed to erase the memory in the situation shown in Fig. 5 is of the order of ∼ 1/J. Furthermore, the adiabatic launching could be exploited to generate specific target states, although we stress that this strategy is inevitably affected by the micro-motion dictated by the opera-torK(t f ) in Eq. (15). Indeed, if one initially prepares the system in the ground state of the static HamiltonianĤ 0 , and then smoothly ramp up the modulation, the system will eventually oscillate around the (target) ground state of the effective Hamilto-nianĤ eff . These micro-motion oscillations may be problematic for specific applications, where the stability of the target state particularly matters; we remind that the micro-motion is inherent to the physics of driven systems, and although it is generally limited in real space, it is typically large in momentum or spin space for the various examples treated in this work [Figs. 4 and 6]. In view of applications, this also highlights the difference between two possible targets: (1) reaching a specific state [e.g. the groundstate ofĤ eff ], or (2) engineering an effective band structure [also associated withĤ eff ]. This aspect of periodically driven systems is left as an open problem for future works.
Finally, we note that the adiabatic launching is not unique, in the sense that the ramping function λ(t) can take arbitrarily many different forms. In particular, one could consider a "sequence preserving adiabatic" protocol where the function λ(t) remains constant during each primitive pulse sequence {Ĥ 0 + V 1 , . . . ,Ĥ 0 +V N } [Eq. (27)]. Let us briefly analyze the timeevolution of such a driving scheme, based on the simple 2-step sequence γ + = {Ĥ 0 +V ,Ĥ 0 −V }, where the + index indicates that the sequence starts with the pulse +V . Suppose that the system is initially prepared in the ground state ψ 0 of the static Hamilto-nianĤ 0 . At the end of the sequence-preserving ramping process, the state has evolved into ψ 0 → ψ + , which is an eigenstate of the effective HamiltonianĤ + associated with the primitive sequence γ + , namelŷ To estimate the micro-motion undergone by the target state ψ + after the ramp, we compute its evolution after half a period. We find Now suppose that the same driving scheme is performed, but using the alternative sequence γ − = {Ĥ 0 −V ,Ĥ 0 +V }. In this case, the target state after the ramp ψ 0 → ψ − satisfieŝ If the eigenvalue E is not degenerate, i.e. ψ micro = ψ − , we find that the target states associated with the two sequences γ ± are sent into each other ψ + ↔ ψ − through the micro-motion [Eqs. (71)- (72)].
If the eigenvalue E is degenerate, which is the case for the ground level of spin-orbit-coupled Hamiltonians [12,13], the evolution is non-trivial and should be studied on a case-by-case basis. In both situations, this analysis further highlights the relevance of the micro-motion in modulated systems.

C. Conclusions
This work was dedicated to the physics of periodically modulated quantum systems, with a view to realizing gauge structures in a wide range of physical contexts.
Our approach was based on the perturbative formalism introduced in Ref. [57], which clearly highlights three relevant notions associated with the driving: the initial kick captured by the op-eratorK(t i ), the effective HamiltonianĤ eff that rules the longtime dynamics, and the micro-motion described byK(t f ) [Section II A]. Based on this perturbative method, we have obtained general formulas and identified diverse driving schemes leading to "non-trivial" effective Hamiltonians, whose characteristics could be useful for the quantum simulation of gauge structures and topological order. In particular, we have discussed the convergence of the perturbative expansion; building on specific examples, we have also presented methods allowing for the partial resummation of infinite series.
This work addresses the general situation where the driving frequency ω is off-resonant with respect to any energy separation present in the problem. However, the effective-Hamiltonian method presented here can also be generalized to describe schemes based on resonant driving [76], as recently implemented to generate synthetic magnetic fields in optical lattices [31][32][33].
We have also mentioned the possibility to switch on the modulations adiabatically; this minimizes the effects attributed to the initial kickK(t i ), which was shown to be considerable when launching the driving abruptly [ Fig. 5]. Schemes based on adiabatic launching could also be exploited to reach interesting target states.
This works also aimed to highlight the important role played by the micro-motion in periodically driven systems. We have shown that these unavoidable oscillations are typically large in momentum and spin space, for the different examples treated in this work. This is particularly relevant from a detection point of view, noting that various probes are built on (possibly spindependent) momentum-distribution imaging. Although these results were obtained in the non-interacting regime, they also suggest that dissipation due to inter-particle collisions could lead to significant heating in cold-matter systems presenting large micromotion. This could be particularly problematic in spin systems, where spin-dependent micro-motion could possibly lead to drastic collision processes. The thermodynamics of driven quantum systems has been recently investigated in Refs. [77][78][79][80] [see also [81,82]]. We note that interactions could also be modulated in cold-atom systems, using time-dependent magnetic fields in a Feshbach resonance [83,84].
Finally, we point out that probing interesting effects in coldmatter systems, such as topological order, generally requires to act on the system with additional potentialsV probe . For instance, measuring the topologically-invariant Chern number in quantum-Hall atomic systems [85][86][87] could be realized by acting on the cloud with an external forceV probe ∼x [see also Ref. [88]]. These additional potentials will contribute to the static Hamilto-nianĤ 0 →Ĥ 0 +V probe , and hence, they will potentially alter the effective Hamiltonian [Eq. (16)] and the corresponding band structure: "measuring the topological order associated with an effective Hamiltonian may destroy it". This phenomenon will be particularly pronounced whenV probe includes Pauli matrices. More generally, adding terms to the static Hamiltonian, either to probe interesting characteristics of the system, or to enrich its topological features, should be handled with care. This issue, which is particularly relevant for the field of quantum simulation, is left as an avenue for future works.
whereH(t) is defined in Eq. (A3). The zeroth-order term was given in Eq. (A4), and the first-order corrections read The evolution operator including first-order corrections is finally given by [Eqs. (A7)-(A8)] where we note that the corrections are small γ 1 for ω 0 ω.
Finally, we show how the formalism of Appendix C allows one to recover the result in Eq. (A9). To second order in (1/ω), the effective Hamiltonian and kick operators read [Eqs. (C10)-(C11)] where we used the fact that V (j) = V (−j) =V /2 in the singleharmonic caseĤ(t) =Ĥ 0 +V cos(ωt). The commutators in Eq. (A10) are readily computed using the operators defined in Eq. (A1), which yieldŝ where Ω and γ are defined in Eqs. (A4)-(A8). Using Eq. (C5), we obtain the evolution operator after one period such that we recover the result in Eq. (A9).

Appendix B: Renormalization of the hopping in modulated optical lattices
In this Appendix, we consider a modulated 1D optical lattice, described by the single-particle Hamiltonian where the periodic function x 0 (t) = x 0 (t + T ) is considered to have a zero mean value over one period T = 2π/ω, and where V OL (x) is the optical lattice potential. In the absence of driving, the static HamiltonianĤ 0 is written in the form of a secondquantized tight-binding Hamiltonian, where the operatorâ † j creates a particle at lattice site x = ja, a is the lattice spacing and J is the hopping matrix element. The modulated lattice is generally studied in a moving frame, in which case the driving acts through an inertial force, where ξ(t) = (ma/ω)ẍ 0 (t). In the following, we set so that the Hamiltonian readŝ Note that the parameter κ introduced in Eq. (4) [main text] is given by κ = ωξ 0 . We also introduce the operator which, together withĤ 0 andV , form a close set under the action of the commutator The latter relations lead to the useful formula where we used Eq. (B8). Finally, the operator evolution after one period is given bŷ where we recover the renormalization of the hopping by the Bessel function of the first kind cos (x sin(τ )) dτ.
We emphasize that the effective Hamiltonian in Eq. (B9) is exact. and consider the unitary transformation The new state φ(t) satisfies the Schrödinger equation where we introduced the effective HamiltonianĤ eff . The method of Ref. [57] then consists in constructing the time-independent effective HamiltonianĤ eff , by transferring all undesired (timedependent) terms into the operatorK(t). The latter has a simple interpretation, which becomes obvious when writing the evolution operator This expression indicates that the evolution can be split into three parts: (a) an initial kick associated with the operatorK(t i ), (b) the evolution dictated by the time-independent effective Hamiltonian H eff , and (c) a final kick associated with the operatorK(t f ).
In general, it is not possible to give an analytical expression for the operatorsĤ eff andK(t) in Eq. (C4) [see however Appendix E for an exactly solvable example]. Thus, it is convenient to build these operators perturbatively, by expanding them in powers of the driving period T = 2π/ω, which is assumed to be small in the problem. Following Ref. [57], we writê and consider the expansions to determine the operatorsĤ eff andK(t) at the desired order O(1/ω n ). Note that we further impose thatK(t) should be pe-riodicK(t) =K(t + T ), with zero mean value over one period. We now apply this strategy to the general situation in Eq. (13), where the HamiltonianĤ(t) of the driven system is given bŷ Following the expansion procedure (C6)-(C8) up to second order O(1/ω 2 ), we obtain the general expressions for the effective Hamiltonian [91] H eff =Ĥ 0 + 1 ω and for the kick operator at time t To derive Eq. (D2), we used the formula which is valid for −π ≤ x ≤ π, see [89].
with N an arbitrary integer. We write the Hamiltonian aŝ where we used the Fourier series in Eq. (28). Applying Eqs. (16) and (17), and presently restricting ourselves to the first-order terms, yieldŝ These expressions can be simplified using the formulas [89] ∞ k=1 sin kx yielding the first-order terms presented in Eqs. (31). A similar calculation allows to evaluate the effective Hamiltonian's second-order terms. For the sake of simplicity, we restrict ourselves to the second-order term Inserting these coefficients into Eq. (F6) yields the final result presented in Eq. (31).
where we note the absence of commutators [V 1 ,V 3 ] and [V 2 ,V 4 ] inĤ eff , and where the commutatorsĈ m,n are defined in Eq. (F7). As for the case N = 3, a proper choice of the operatorsĤ 0 and V 1,2,3,4 can potentially lead to non-trivial effects that are first order in (1/ω).
Let us now focus on the α sequence in Eq. (32), introduced in Section V A.
First, we note that the third order corrections are easily obtained when the α sequence in Eq. (32) is approximated by the smooth single-harmonic drivingV (t) =Â cos(ωt) +B sin(ωt). Pushing the perturbative expansion of Appendix C to the next order, we find where we note that π 3 /768 ≈ 1/16, in agreement with Eq. (G3). Note that this method disregards the terms at this order of the computation; indeed, the alternative approach implicitly assumes that these terms will contribute to higher orders in 1/ω [Appendix K]. However, we note that the expression in Eq. (G4) applies to the SOC scheme analyzed in Section VII B 1, for which the neglected terms in Eq. (G5) all vanish identically. The effects associated with the third order corrections (G4) are illustrated for this specific scheme in Fig. 4.

Symbol
Tight-binding operator Moreover, it is useful to note the cyclic conditions [see also Eq.